On the (3,1)-choosability of planar graphs without adjacent cycles of length 5,6,7

A $(k,d)$-list assignment $L$ of a graph $G$ is a mapping that assigns to each vertex $v$ a list $L\left(v\right)$ of at least $k$ colors satisfying $|L\left(x\right)\cap L\left(y\right)|\le d$ for each edge $xy$. A graph $G$ is $(k,d)$-choosable if there exists an $L$-coloring of $G$ for every $(k,d)$-list assignment $L$. This concept is also known as choosability with separation. In this paper, we prove that any planar graph $G$ is $(3,1)$-choosable if any $i$-cycle is not adjacent to a $j$-cycle, where $5\le i\le 6$ and $5\le j\le 7$.